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   http://functions.wolfram.com/13.06.06.0006.01
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    EulerPhi[p] == Sum[DivisorProduct[1 - (1/Subscript[d, j]) 
      Sum[Exp[(2 Pi I j k)/Subscript[d, j]], {k, 0, Subscript[d, j] - 1}], 
    {Subscript[d, j], p}], {j, 1, p - 1}] /; 
 Element[p, Primes] && Element[Subscript[d, j], Divisors[n]] 
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   Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EulerPhi", "[", "p", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], RowBox[List["DivisorProduct", "[", RowBox[List[RowBox[List["1", "-", RowBox[List[FractionBox["1", SubscriptBox["d", "j"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[SubscriptBox["d", "j"], "-", "1"]]], RowBox[List["Exp", "[", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "j", " ", "k"]], SubscriptBox["d", "j"]], "]"]]]]]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "p"]], "}"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Primes"]], "\[And]", RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", "n", "]"]]]]]]]]]] 
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   <math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'>  <semantics>  <mrow>  <mrow>  <mrow>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", EulerPhi] </annotation>  </semantics>  <mo> ( </mo>  <mi> n </mi>  <mo> ) </mo>  </mrow>  <mo> ⩵ </mo>  <mstyle scriptlevel='0'>  <mrow>  <munderover>  <mo> ∑ </mo>  <mrow>  <mi> j </mi>  <mo> = </mo>  <mn> 1 </mn>  </mrow>  <mrow>  <mi> p </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </munderover>  <mrow>  <munder>  <mo> ∏ </mo>  <mrow>  <msub>  <mi> d </mi>  <mi> j </mi>  </msub>  <mtext>   </mtext>  <mo> | </mo>  <mtext>   </mtext>  <mi> p </mi>  </mrow>  </munder>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mrow>  <mfrac>  <mn> 1 </mn>  <msub>  <mi> d </mi>  <mi> j </mi>  </msub>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <munderover>  <mo> ∑ </mo>  <mrow>  <mi> k </mi>  <mo> = </mo>  <mn> 0 </mn>  </mrow>  <mrow>  <msub>  <mi> d </mi>  <mi> j </mi>  </msub>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </munderover>  <mrow>  <mi> exp </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mfrac>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> j </mi>  <mo> ⁢ </mo>  <mi> k </mi>  </mrow>  <msub>  <mi> d </mi>  <mi> j </mi>  </msub>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </mstyle>  </mrow>  <mo> /; </mo>  <mrow>  <mrow>  <mi> p </mi>  <mo> ∈ </mo>  <semantics>  <mi> ℙ </mi>  <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalP]", Function[Primes]] </annotation>  </semantics>  </mrow>  <mo> ∧ </mo>  <mrow>  <msub>  <mi> d </mi>  <mi> j </mi>  </msub>  <mo> ∈ </mo>  <mrow>  <mi> divisors </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> n </mi>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <ci> FormBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <ms> ϕ </ms>  <ci> EulerPhi </ci>  </apply>  <ms> ( </ms>  <ms> n </ms>  <ms> ) </ms>  </list>  </apply>  <ms> ⩵ </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> UnderoverscriptBox </ci>  <ms> ∑ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> j </ms>  <ms> = </ms>  <ms> 1 </ms>  </list>  </apply>  <apply>  <ci> RowBox </ci>  <list>  <ms> p </ms>  <ms> - </ms>  <ms> 1 </ms>  </list>  </apply>  </apply>  <apply>  <ci> ErrorBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> UnderscriptBox </ci>  <ms> ∏ </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms> d </ms>  <ms> j </ms>  </apply>  <ms> | </ms>  <ms> p </ms>  </list>  </apply>  </apply>  <apply>  <ci> RowBox </ci>  <list>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> 1 </ms>  <ms> - </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> FractionBox </ci>  <ms> 1 </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> d </ms>  <ms> j </ms>  </apply>  </apply>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> UnderoverscriptBox </ci>  <ms> ∑ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> k </ms>  <ms> = </ms>  <ms> 0 </ms>  </list>  </apply>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms> d </ms>  <ms> j </ms>  </apply>  <ms> - </ms>  <ms> 1 </ms>  </list>  </apply>  </apply>  <apply>  <ci> RowBox </ci>  <list>  <ms> exp </ms>  <ms> ( </ms>  <apply>  <ci> FractionBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <ms> 2 </ms>  <ms> π </ms>  <ms> ⅈ </ms>  <ms> j </ms>  <ms> k </ms>  </list>  </apply>  <apply>  <ci> SubscriptBox </ci>  <ms> d </ms>  <ms> j </ms>  </apply>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  </list>  </apply>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  </apply>  </list>  </apply>  </list>  </apply>  <ms> /; </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <ms> p </ms>  <ms> ∈ </ms>  <apply>  <ci> TagBox </ci>  <ms> ℙ </ms>  <apply>  <ci> Function </ci>  <primes />  </apply>  </apply>  </list>  </apply>  <ms> ∧ </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms> d </ms>  <ms> j </ms>  </apply>  <ms> ∈ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> divisors </ms>  <ms> ( </ms>  <ms> n </ms>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  </list>  </apply>  </list>  </apply>  <ci> TraditionalForm </ci>  </apply>  </annotation-xml>  </semantics>  </math> 
   
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  | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EulerPhi", "[", "p_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], RowBox[List["DivisorProduct", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[SubscriptBox["d", "j"], "-", "1"]]], SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "j", " ", "k"]], SubscriptBox["d", "j"]]]]], SubscriptBox["d", "j"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "p"]], "}"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Primes"]], "&&", RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", "n", "]"]]]]]]]]]]]]  |  
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   Date Added to functions.wolfram.com (modification date)
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